Linear Perturbation Analysis Research Articles

Using kinetic theory to examine a self-gravitating system composed of baryons and cold dark matter

We examine the evolution of non-relativistic cold dark matter gravitationally coupled to baryons with modes deep inside the Hubble radius (sub-horizon regime) using a kinetic theory approach within the realm of Newtonian theory. We obtain the general solution for the total density perturbation and we also show that a baryon perturbation catches up with the dark matter perturbation at late times, which in turn makes possible the formation of bound structures. We extend the linear perturbation analysis by considering the turn-around event, collapse of matter, and its virialization process.

Buckling initiation in layered hydrogels during transient swelling

Subjected to compressive stresses, soft polymers with stiffness gradients can display various buckling patterns. These compressive stresses can have different origins, like mechanical forces, temperature changes, or, for hydrogel materials, osmotic swelling. Here, we focus on the influence of the transient nature of osmotic swelling on the initiation of buckling in confined layered hydrogel structures. A constitutive model for transient hydrogel swelling is outlined and implemented as a user-subroutine for the commercial finite element software Abaqus. The finite element procedure is benchmarked against linear perturbation analysis results for equilibrium swelling showing excellent correspondence. Based on the finite element results we conclude that the initiation of buckling in a two-layered hydrogel structure is highly affected by transient swelling effects, with instability emerging at lower swelling ratios and later in time with a lower diffusion coefficient. In addition, for hard-on-soft systems the wavelength of the buckling pattern is found to decrease as the diffusivity of the material is reduced for gels with a relatively low stiffness gradient between the substrate and the upper film. This study highlights the difference between equilibrium and transient swelling when it comes to the onset of instability in hydrogels, which is believed to be of importance as a fundamental aspect of swelling as well as providing input to guiding principles in the design of specific hydrogel systems.

Low-frequency selectivity in flat-plate boundary layer with elliptic leading edge

Linear perturbation analyses of zero-pressure-gradient boundary layers at subcritical Reynolds numbers predict that transient disturbance amplification can take place due to the lift-up mechanism. Upstream, streamwise-elongated vortices yield the largest response per unit of inflow disturbance energy, which takes the form of streamwise-elongated streaks. In this work, we compute the linear and also nonlinear inflow disturbances that generate the largest response inside the boundary layer, for flow over a thin flat plate with a slender leading edge. In order to compare our results with earlier linear analyses, we constrain the inlet disturbance to be monochromatic in time, or a single frequency. The boundary layer effectively filters high frequencies, and only low-frequency perturbations induce a strong response downstream. The low-frequency optimal inflow disturbance has a spanwise wavenumber that scales with $\sqrt{Re}$, and it consists of streamwise and normal vorticity components: the latter is tilted around the leading edge into the streamwise direction and, further downstream, generates streaks. While none of the computed monochromatic disturbances alone can lead to breakdown to turbulence, secondary instability analyses demonstrate that the streaky base state is unstable. Nonlinear simulations where the inflow disturbance is supplemented with additional white noise undergo secondary instability and breakdown to turbulence.

Is the local Hubble flow consistent with concordance cosmology?

Yes. In a perturbed Friedmann model, the difference of the Hubble constants measured in two rest-frames is independent of the source peculiar velocity and depends only on the relative velocity of the observers, to lowest order in velocity. Therefore this difference should be zero when averaging over sufficient sources, which are at large enough distances to suppress local nonlinear inhomogeneity. We use a linear perturbative analysis to predict the Doppler effects on redshifts and distances. Since the observed redshifts encode the effect of local bulk flow due to nonlinear structure, our linear analysis is able to capture aspects of the nonlinear behaviour. Using the largest available distance compilation from CosmicFlows-3, we find that the data is consistent with simulations based on the concordance model, for sources at 20−150 Mpc.

Topology optimization for eigenfrequencies of a rotating thin plate via moving morphable components

An efficient and explicit topology optimization approach is initially proposed for eigenfrequencies of a rotating thin plate in this study. First of all, an accurate dynamic model of the rotating thin plate is established via the thin plate elements of the absolute nodal coordinate formulation (ANCF). When performing the modal characteristic analysis of the rotating thin plate at a prescribed angular velocity, the linear perturbation analysis is employed, during which the coupling between the membrane and bending deformations is considered. The coupling term makes the dynamic model established in the study more accurate than conventional models, especially in the case of large deformations. Then, the moving morphable components (MMC) are used to describe the topology of the plate. In the frame of MMC-based topology optimization, explicit geometrical parameters, positions, and orientations of the components are taken as the design variables so that the total number of the design variables can be greatly reduced. For the topology optimization, the sensitivities of a simple eigenfrequency and multiply repeated eigenfrequencies with respect to a design variable are analytically derived. During the optimization, in order to remove the localized modes in the low-density areas, the mass and stiffness matrices of the thin plate elements of ANCF are carefully penalized. Finally, four numerical examples are presented to validate the proposed topology optimization approach and to demonstrate its effectiveness for two objectives, i.e., maximizing either the first eigenfrequency or the gap between two consecutive eigenfrequencies of a rotating thin plate.

Strategies to cure numerical shock instability in the HLLEM Riemann solver

SummaryThe HLLEM scheme is a popular contact and shear preserving approximate Riemann solver that is known to be plagued by various forms of numerical shock instability. In this paper, we clarify that the shock instability exhibited by this scheme is primarily triggered by the spurious activation of the antidiffusive terms present in the first and third Riemann flux components on the transverse interfaces adjoining the shock front due to numerical perturbations. These erroneously activated terms are shown to counteract the favorable damping mechanism provided by its inherent HLL‐type diffusive terms, causing an unphysical variation of the conserved quantity ρu both along and across the numerical shock. To prevent this, two distinct strategies are proposed termed as Selective Wave Modification and Anti Diffusion Control. The former focuses on enhancing the quantity of the favorable HLL‐type dissipation available on these critical flux components by carefully increasing the magnitudes of certain nonlinear wave speed estimates, while the latter focuses on directly controlling the magnitude of these critical antidiffusive terms. A linear perturbation analysis is performed to gauge the effectiveness of these cures and to estimate a von Neumann–type stability bounds on the CFL number associated with their use. Results from a variety of classic shock instability test cases show that the proposed strategies are able to provide excellent shock stable solutions even on grids that are highly elongated across the shock front without compromising the accuracy on inviscid contact or shear dominated viscous flows.

On the Morphodynamic Equilibrium of a Short Tidal Channel

We set up a two‐dimensional modeling framework to explore the development and the equilibrium configuration of a tidal channel. We consider two types of morphodynamic modeling systems coupling different flow and sedimentation models, to study the feedback mechanisms controlling the morphodynamics of tidal channels. The simulations indicate that the tidal channel first experiences a pseudo equilibrium state, then diverges from it, and finally approaches a dynamic equilibrium. The reason for this behavior is demonstrated analytically using linear perturbation analysis. We identify distinct positive and negative feedbacks that lead the tidal system to a dynamic equilibrium. One of the feedback mechanisms is controlled by the flow momentum balance, while the other one depends on the overall sediment balance. The dynamic equilibrium configurations obtained in the numerical simulations compare favorably with the theoretical equilibrium. Deviations from theory can be explained analyzing the assumptions in the flow and sedimentation models. Moreover, the minimum width‐to‐depth relationship of a tidal channel cross section is derived and is found to be in agreement with field measurements.

Metal Electrode Surfaces Can Roughen Despite the Constraint of a Stiff Electrolyte

Roughening of metal electrodes in batteries is detrimental as it can lead to metal dendrites. Such dendrites can cause short circuits when they grow from the metal electrode to the other one, as can happen during battery operation when metal is plated onto the surface of an electrode. It has been suggested that solid electrolytes of sufficient elastic stiffness can suppress electrode surface roughening and dendriting, although experimental evidence is now emerging that this possibility is not valid. To investigate whether metal electrode surfaces will roughen during battery charging we carry out a linear perturbation analysis. Our calculations explore whether an electrode surface with one-dimensional sinusoidal roughness will experience growth of its amplitude. We assess a linear elastic electrolyte that is a single ion conductor bonded to a metal electrode being plated by a cathodic ionic current. We find that long wavelength perturbations will always increase in roughness. High current densities during battery charging are found to permit growth of the amplitude of small wavelength roughness. The stiffness of the solid electrolyte is found to play a role in limiting the growth of roughness, but its effect can always be overcome at high current densities and for long wavelength protrusions.

Linear perturbations of low angular momentum accretion flow in the Kerr metric and the corresponding emergent gravity phenomena

For certain geometric configuration of matter falling onto a rotating black hole, we develop a novel linear perturbation analysis scheme to perform the stability analysis of stationary integral accretion solutions corresponding to the steady state low angular momentum, inviscid, barotropic, irrotational, general relativistic accretion of hydrodynamic fluid. We demonstrate that such steady states remain stable under linear perturbation, and hence the stationary solutions are reliable to probe the black hole spacetime using the accretion phenomena.We report that a relativistic acoustic geometry emerges out as the consequence of such stability analysis procedure. We study various properties of that sonic geometry in detail. We construct the causal structures to establish the one to one correspondences of the sonic points with the acoustic black hole horizons, and the shock location with an acoustic white hole horizon. The influence of the spin of the rotating black holes on the emergence of such acoustic spacetime has been discussed.

Physical objects approaching the Cauchy horizon of a rapidly rotating Kerr black hole

We solve the 2+1-dimensional Teukolsky equation numerically for the Weyl scalars $\psi_0$ and $\psi_4$ along a time-like geodesic approaching the Cauchy horizon of a rapidly rotating perturbed Kerr black hole. We find that both the amplitude and frequency of the Weyl scalars agree with the results of linear perturbation analysis. We then model a physical object by a simple damped harmonic oscillator, which is driven by an external force that mimics the tidal force experienced by the infalling object. We use this model to find the total deformation of the object at the Cauchy horizon, and the resonant effect when the driving force's frequency matches the internal frequency of the oscillator that models the object.

A simple cure for numerical shock instability in the HLLC Riemann solver

The Harten–Lax–van Leer with contact (HLLC) approximate Riemann solver is known to be plagued by various forms of numerical shock instability. In this paper, we propose a new framework for developing shock stable versions of the HLLC scheme for the Euler system of equations without compromising on its contact and shear preserving ability. The proposed framework, termed as Selective Wave Modification (SWM), identifies and enhances the magnitude of the inherent HLL-type diffusive component within the HLLC scheme in the vicinity of a normal shock front. This is achieved by suitably increasing the magnitudes of the nonlinear wave speed estimates appearing in them through a dissipation parameter ϵ. A linear perturbation analysis is performed to gauge the effectiveness of the proposed framework in damping unwanted perturbations in physical quantities and is also used to estimate a von-Neumann-type stability bound on the CFL number associated with its use. Two distinct methods to estimate ϵ, which results in HLLC-SWM-E (Eigenvalue based) and HLLC-SWM-P (Pressure based) schemes, are proposed and compared. Further, a matrix based stability analysis is used to show that the proposed schemes can be configured to remain shock stable over a wide range of free stream Mach numbers. Numerical results confirm that the proposed schemes are capable of computing shock stable solutions for a wide variety of problems. On viscous flows, the HLLC-SWM-P variant is found to be the most accurate. Performance of these schemes suggests that it is possible to construct shock stable upwind schemes that simultaneously retains accuracy on the linearly degenerate wavefields.

Stealth Schwarzschild solution in shift symmetry breaking theories

We find stealth Schwarzschild solutions with a nontrivial profile of the scalar field regular on the horizon in the Einstein gravity coupled to the scalar field with the k-essence and/or generalized cubic galileon terms, which is a subclass of the Horndeski theory breaking the shift symmetry, where the propagation speed of gravitational waves coincides with the speed of light. After deriving sufficient conditions for the shift symmetry breaking theory to allow a general Ricci-flat metric solution with a nontrivial scalar field profile, we focus on the stealth Schwarzschild solution with the scalar field with or without time dependence. For the profile $\phi=\phi_0(r)$, we explicitly obtain two types of stealth Schwarzschild solutions, one of which is regular on the event horizon. The linear perturbation analysis clarifies that the kinetic term of the scalar mode identically vanishes, indicating that the scalar mode is strongly coupled. The absence of the kinetic term of the scalar mode in the quadratic action would inevitably arise for the stealth Schwarzschild solutions in the theory with a general scalar field profile depending only on the spatial coordinates. On the other hand, for the time-dependent scalar field profile, we clarify that there does not exist a stealth Schwarzschild solution in the shift symmetry breaking theories.

Modal analysis of historical masonry structures: Linear perturbation and software benchmarking

The mechanical behavior of masonry materials has a common feature: a nonlinear behavior with high compressive strength and very low tensile strength. As a consequence, old masonry buildings generally present cracks due to permanent loads and/or accidental events. Therefore, the characterization of the global dynamic behavior of masonry structures should take into account the presence of existing cracks. This paper presents a numerical approach coupling linear perturbation and modal analysis in order to estimate the dynamic properties of masonry constructions, taking into account the existence of structural damage. First, the approach is validated on a masonry arch subjected to increasing loads, via three FE codes. Then, the same procedure is applied to a real masonry structure affected by a severe crack distribution.

Wrinkling and ratcheting of a thin film on cyclically deforming plastic substrate: Mechanical instability of the solid-electrolyte interphase in Li–ion batteries

Lithium–ion batteries (LIBs) in consumer electronics, electric cars and large-scale energy storage systems are often susceptible to capacity fading due to mechanical degradation of the solid-electrolyte interphase (SEI) layer on the electrodes. Here we present an analytical model to describe SEI wrinkling and ratcheting behaviors during cyclic lithiation and delithiation of LIBs. The SEI-electrode system is modeled as a bilayer structure consisting of a thin film resting on a plastic substrate. Surface instability is found in such a system under cyclic plastic deformation induced by lithiation and delithiation. A linear perturbation analysis is performed to determine the critical wrinkling strain and wavenumber. The interfacial shear traction induced by surface wrinkling can further lead to plastic ratcheting, and the wrinkling amplitude increases with each lithiation/delithiation cycle. A phase diagram is plotted to characterize and predict different system behaviors, e.g., elastic, elastic wrinkling, shakedown without wrinkling, shakedown with wrinkling, and ratcheting. A series of finite element simulations are performed to validate the theoretical predictions. The analysis suggests that the mechanical instabilities of the SEI, including wrinkling and ratcheting, can be prevented by several strategies, such as introducing an artificial SEI with a sufficiently large stiffness and thickness, and/or with a tensile pre-stress in the SEI.

Determining the stiffness of the equation of state using low T/W dynamical instabilities in differentially rotating stars

We investigate the nature of low T/W dynamical instabilities in various ranges of the stiffness of the equation of state in differentially rotating stars. Here T is the rotational kinetic energy, while W the gravitational binding energy. We analyze these instabilities in both a linear perturbation analysis and a three-dimensional hydrodynamical simulation. An unstable normal mode of a differentially rotating star is detected by solving an eigenvalue problem along the equatorial plane of the star. The physical mechanism of low T/W dynamical instabilities is also qualitatively confirmed by a scattering of sound waves between corotation and the surface caused by the corotation barrier. Therefore, we can draw a picture of existing pulsation modes unstabilized due to an amplified reflection of sound waves from the corotation barrier. The feature in the eigenfrequency and eigenfunction of the unstable mode in the linear analysis roughly agrees with that in the three-dimensional hydrodynamical simulation in Newtonian gravity. Moreover, the nature of the eigenfunction that oscillates between corotation and the surface for an unstable star requires reinterpretation of pulsation modes in differentially rotating stars. Finally, we propose a manner by which to constrain the stiffness of the equation of state by the direct detection of mode decomposed gravitational waveforms.

A cure for numerical shock instability in HLLC Riemann solver using antidiffusion control

Various forms of numerical shock instabilities are known to plague many contact and shear preserving approximate Riemann solvers, including the popular Harten-Lax-van Leer with Contact (HLLC) scheme, during high speed flow simulations governed by the Euler system of equations. In this paper we propose a simple and inexpensive novel strategy to prevent the HLLC scheme from developing such spurious solutions without compromising on its linear wave resolution ability. The cure is primarily based on a reinterpretation of the HLLC scheme as a combination of its well-known diffusive counterpart, the HLL scheme, and an antidiffusive term responsible for its accuracy on linear wavefields. In our study, a linear analysis of this alternate form indicates that shock instability in the HLLC scheme could be triggered due to the unwanted activation of the antidiffusive term appearing in its mass and interface-normal momentum flux component discretizations on interfaces that are not aligned with the shock front. This inadvertent activation results in weakening of the favourable dissipation provided by its inherent HLL scheme and causes unphysical mass flux variations along the shock front. To mitigate this, we propose a modified HLLC scheme that employs a simple differentiable pressure-ratio based multidimensional shock sensor to achieve smooth control of these critical antidiffusive terms near shocks. Using a linear perturbation analysis and a matrix based stability analysis, we establish that the resulting scheme, called HLLC-ADC (Anti-Diffusion Control), is shock stable over a wide range of freestream Mach numbers. Results from standard numerical test cases demonstrate that the HLLC-ADC scheme is indeed free from the most common manifestations of shock instability including the Carbuncle phenomenon without significant loss of accuracy on shear dominated viscous flows.

The Gravitational Wave Signal from Core-collapse Supernovae

Abstract We study gravitational waves (GWs) from a set of 2D multigroup neutrino radiation hydrodynamic simulations of core-collapse supernovae (CCSNe). Our goal is to systematize the current knowledge about the post-bounce CCSN GW signal and recognize the templatable features that could be used by the ground-based laser interferometers. We demonstrate that, starting from ∼400 ms after core bounce, the dominant GW signal represents the fundamental quadrupole (l = 2) oscillation mode (f-mode) of the proto–neutron star (PNS), which can be accurately reproduced by a linear perturbation analysis of the angle-averaged PNS profile. Before that, in the time interval between ∼200 and ∼400 ms after bounce, the dominant mode has two radial nodes and represents a g-mode. We associate the high-frequency noise in the GW spectrograms above the main signal with p-modes, while below the dominant frequency there is a region with very little power. The collection of models presented here summarizes the dependence of the CCSN GW signal on the progenitor mass, equation of state, many-body corrections to the neutrino opacity, and rotation. Weak dependence of the dominant GW frequency on the progenitor mass motivates us to provide a simple fit for it as a function of time, which can be used as a prior when looking for CCSN candidates in the LIGO data.

Nonlinear spherical perturbations in quintessence models of dark energy

Observations have confirmed the accelerated expansion of the universe. The accelerated expansion can be modelled by invoking a cosmological constant or a dynamical model of dark energy. A key difference between these models is that the equation of state parameter w for dark energy differs from −1 in dynamical dark energy (DDE) models. Further, the equation of state parameter is not constant for a general DDE model. Such differences can be probed using the variation of scale factor with time by measuring distances. Another significant difference between the cosmological constant and DDE models is that the latter must cluster. Linear perturbation analysis indicates that perturbations in quintessence models of dark energy do not grow to have a significant amplitude at small length scales. In this paper we study the response of quintessence dark energy to non-linear perturbations in dark matter. We use a fully relativistic model for spherically symmetric perturbations. In this study we focus on thawing models. We find that in response to non-linear perturbations in dark matter, dark energy perturbations grow at a faster rate than expected in linear perturbation theory. We find that dark energy perturbation remains localised and does not diffuse out to larger scales. The dominant drivers of the evolution of dark energy perturbations are the local Hubble flow and a supression of gradients of the scalar field. We also find that the equation of state parameter w changes in response to perturbations in dark matter such that it also becomes a function of position. The variation of w in space is correlated with density contrast for matter. Variation of w and perturbations in dark energy are more pronounced in response to large scale perturbations in matter while the dependence on the amplitude of matter perturbations is much weaker.

The local stability of the magnetized advection-dominated discs with the radial viscous force

We study local stability of the advection-dominated optically thick (slim) and optically thin discs with purely toroidal magnetic field and the radial viscous force using a linear perturbation analysis. Our dispersion relation indicates that the presence of magnetic fields and radial viscous force cannot give rise to any new mode of the instability. We find, however, that growth rate of the thermal mode in the slim discs and that of the acoustic modes in the slim and optically thin discs are dramatically affected by the radial viscous force. This force tends to strongly decrease the growth rate of the outward-propagating acoustic mode (O-mode) at the short-wavelength limit, but it causes a slim disc to become thermally more unstable. We find that growth rate of the thermal mode increases in the presence of radial viscous force. This enhancement is more significant when the viscosity parameter is large. We also show that growth rate of the O-mode reduces when radial viscous force is considered. The growth rates of the thermal and acoustic modes depend on the magnetic field. Although the instability of O-mode for a stronger magnetic field case has a higher growth rate, the thermal mode of the slim discs can be suppressed when the magnetic field is strong. The inertial-acoustic instability of a magnetized disc may explain the quasi-periodic oscillations (QPOs) from the black holes.

Electrohydrodynamic stability of a plasma-liquid interface

Many plasma applications involve the plasma coming into contact with a liquid surface. Previous analyses of the stability of such liquid surfaces have neglected the presence of the sheath region between the bulk plasma and the liquid. Large electric fields, typically in excess of several MV m−1, and strong ion flows are present in this region. This paper considers a linear perturbation analysis of a liquid-sheath interface in order to find the marginal condition for instability. This condition shows that molten metal surfaces in tokamak edge plasmas are stable against the electric field, if a normal sheath is formed, due to the impact of ions on the surface. The stabilization of the liquid surface by ion bombardment is encouraging for the ongoing development of plasma-liquid technologies.